Hostname: page-component-76c49bb84f-7l7zb Total loading time: 0 Render date: 2025-07-09T21:15:29.068Z Has data issue: false hasContentIssue false
  • English
  • Français

Some hyperbolicity revisited and robust transitivity

Part of: Dynamical systems with hyperbolic behavior Topological dynamics

Published online by Cambridge University Press:  23 June 2025

LUIS PEDRO PIÑEYRÚA
LUIS PEDRO PIÑEYRÚA*
Affiliation:
IMERL, Facultad de Ingeniería, Universidad de la República, Montevideo 11300, Montevideo, Uruguay
*
e-mail: lpineyrua@fing.edu.uy
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article, we revisit the notion of some hyperbolicity introduced by Pujals and Sambarino [A sufficient condition for robustly minimal foliations. Ergod. Th. & Dynam. Sys. 26(1) (2006), 281–289]. We present a more general definition that, in particular, can be applied to the symplectic context (something that was not possible for the previous one). As an application, we construct $C^1$ robustly transitive derived from Anosov diffeomorphisms with mixed behaviour on centre leaves.

Keywords

some hyperbolicityrobust transitivity

MSC classification

Primary: 37D30: Partially hyperbolic systems and dominated splittings
Secondary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 32
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Anosov, D. V.. Geodesic flows on closed Riemann manifolds with negative curvature. Proc. Steklov Inst. Math. 90 (1967), 1–235; Engl. transl. (American Mathematical Society, Providence, RI, 1969).Google Scholar
Berger, P. and Carrasco, P.. Non-uniformly hyperbolic diffeomorphisms derived from the standard map. Comm. Math. Phys. 329 (2014), 239–262.10.1007/s00220-014-1932-1CrossRefGoogle Scholar
Bonatti, C. and Díaz, L. J.. Persistent nonhyperbolic transitive diffeomorphisms. Ann. of Math. (2) 143 (1996), 357–396.10.2307/2118647CrossRefGoogle Scholar
Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115 (2000), 157–193.10.1007/BF02810585CrossRefGoogle Scholar
Bonk, M. and Kleiner, B.. Rigidity for quasi-Möbius group actions. J. Differential Geom. 61 (2002), 81–106.10.4310/jdg/1090351321CrossRefGoogle Scholar
Carrasco, P. and Obata, D.. A new example of robustly transitive diffeomorphism. Math. Res. Lett. 28(3) (2021), 665–679.10.4310/MRL.2021.v28.n3.a2CrossRefGoogle Scholar
Cerminara, M. and Lewowicz, J.. Some open problems concerning expansive systems. Rend. Istit. Mat. Univ. Trieste 42 (2010), 129–141.Google Scholar
Fisher, T., Potrie, R. and Sambarino, M.. Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov. Math. Z. 278 (2014), 149–168.10.1007/s00209-014-1310-xCrossRefGoogle Scholar
Franks, J.. Anosov diffeomorphisms. Proc. Sympos. Pure Math. 14 (1970), 61–93.10.1090/pspum/014/0271990CrossRefGoogle Scholar
Franks, J.. Necessary conditions for stability of diffeomorphisms. Trans. Amer. Math. Soc. 158(2) (1971), 301–308.10.1090/S0002-9947-1971-0283812-3CrossRefGoogle Scholar
Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer, Berlin, 1977.10.1007/BFb0092042CrossRefGoogle Scholar
Leguil, M. and Zhang, Z.. ${C}^r$ -prevalence of stable ergodicity for a class of partially hyperbolic systems. J. Eur. Math. Soc. (JEMS) 24(9) (2022), 3379–3438.10.4171/jems/1163CrossRefGoogle Scholar
Lewowicz, J.. Lyapunov functions and topological stability. J. Differential Equations 38 (1980), 192–209.10.1016/0022-0396(80)90004-2CrossRefGoogle Scholar
Mañé, R.. Contributions to the stability conjecture. Topology 17(4) (1978), 383–396.10.1016/0040-9383(78)90005-8CrossRefGoogle Scholar
Potrie, R.. Partially hyperbolicity and attracting regions in 3-dimensional manifolds. PhD Thesis, 2012. https://arxiv.org/abs/1207.1822.Google Scholar
Pujals, E. and Sambarino, M.. A sufficient condition for robustly minimal foliations. Ergod. Th. & Dynam. Sys. 26(1) (2006), 281–289.10.1017/S0143385705000568CrossRefGoogle Scholar
Shub, M.. Topological transitive diffeomorphism on ${T}^4$ . Proc. Symp. on Differential Equations and Dynamical Systems (Lecture Notes in Mathematics, 206). Springer, Berlin, 1971, pp. 39–40.10.1007/BFb0070139CrossRefGoogle Scholar